Question

How do you use the limit definition to find the slope of the tangent line to the graph `y=x^3` at (-2,8)?

Answer 1

Slope: `color(green)(12)`
(see below for application of limit definition)

Explanation:

Replacing `y` with `f(x)`
the limit definition of the slope is
`color(white)("XXX")m_x =lim_(hrarr0)(f(x+h)-f(x))/h`

In this case
`color(white)("XXX")m_x=lim_(hrarr0)((x+h)^3-x^3)/h`

`color(white)("XXXX")=lim_(hrarr0)(x^3+3x^2h+3xh^2+h^3-x^3)/h`

`color(white)("XXXX")=lim_(hrarr0)(3x^2h+3xh^2+h^3)/h`

`color(white)("XXXX")=lim_(hrarr0)(3x^2+3xh+h^2)`

`color(white)("XXXX")=3x^2`

At `(-2,8)`
`color(white)("XXX")x=-2`
and
`color(white)("XXX")m_(-2)=3(-2)^2 = 3xx4=12`